1st: After getting the crown and giving it to Haedrig, you can go straight to the cathedral garden and to Leoric's passage, which skips talking to Cain in town.

2nd: You don't have to talk to Kormac the templar after helping him, you can go straight to his stash (forward and left) and he will follow.

3rd: You do not have to talk to the enchanter after freeing her from the 1st spider lady. You can go and find the Khazra staff right away. You may doubt this from the quest marker, but once you find the staff, the quest will shift.

4th: In act 2, after you kill the first cultists, you can run past the guards protecting the people, it will auto-complete.

5th: After going into the sewers in search of Adria, upon entering the wretched pit, there is a very small window that you can enter the area of the snakes and kill some during the cutscene, making it more likely that you will not have to wait for them to uncloak. This is mostly only possible in NG+.

6th: While looking for the crucible, you want to go as right as you possibly can, it will always be to the right (Right being top right of the screen).

7th: In act 3, while going to the first catapult to raise, you do not need to stop to talk to the quest giver, just go straight to the demons at the barricade.

8th: In act 3, after crossing the 2nd bridge and killing the siege beast, you can enter the crater immediately without stopping to talk to Adria.

9th: In act 5, you do not have to stop to talk to Victor at the start before defending the church. Just go past and open the door.

10th: After getting into the Pandemonium Fortress, you don't have to talk to your ghost buddy, you can just go straight right and skip the text.

Most of these were taken from dowolf's NG+ monk run.